In (35), (38), (87)
re presents tlie meau temparature (above tlie staiulard) of tlie solui cyliniler of radius r. It is to be remarked tliat tlie doublé refraction of tlie ray at r is independent of tlie values of 4 bevond r, and also of any boundarv-pressure. If 4 increases (or decreases) continuously froui tlie centre outxvards, tlie doublé refraction liever vaiiishes, and no dark cir-
cle is seen in tlie polariscope.
In the above solution if tlie eylinder is terininated by «at faces. we must imagine suitable forces /.', given by (2b), to be operative over the faces. The iutegral of these forces may be reduced to zero by allowing a suitable expansion parallel to the axis. llegarding dirih as a constant inot necessarily zero), independent ot r and z, we have in place ot ' »^)
*-*@ + 0 + O+*»Z-" (:IS)
The additions to P and Q are /. du-tl:, wliile (P—U) remains unclianged.
If the eylinder is long relatively to its diameter, tlie last state ot tliings mav be supposed to remain approximately unclianged, i\t n tliough the terminal faces be free froni applied force. In the neighbourhood of the ends there will be local disturbances, requiring a more elaborate analysis for tlieir calculation, hut the simple solution will applv to the greater part of the lengtli.
The case of a tliin plate wliose faces are everywliere free froni applied force is more ditticult to tieat in a rigorous manner, hut the tollowing is probably a sufficiënt account of the matter. By supposing /J=0 m (:5 b) we get
and using this value of dtr </:,
» ^ ('<": . ^"<1 (io)
,lr^r) '<lr >.+2y
Q= (''" + "^+2/"- (41)
/, ■-)- iy.\dr 1 rJ r >.-\-~y.